A recent NBPTS report entitled Student Learning, Student Achievement: How Do Teachers Measure Up? included this diagram:
This struck me as quite accurate.
I reflected on an incident in class this morning. My Math 1 students were learning to create a circle of a given area using a compass. They each had an 8-cm circle on their paper when I asked them to draw a circle with half the area. Inevitably, many students cut the radius in half and scribed a circle with 1/4 of the original area.
Student [skeptically]: Is this right?
Me: Does it look like it has half the area?
Student: No - so it's wrong?
Me: Can you find the area?
Student: So it's wrong?
Me: It's not wrong if you learn from it.
Student: So it's wrong [begins erasing].
Even at the time, I realized that the student was learning, even though he may not be able to solve the problem yet. This type of learning falls in the bottom base level of the pyramid above - something identifiable but difficult or impossible to quantify. (The student's preoccupation with having an answer deemed "right" or "wrong" is another topic altogether).
How have you come across unquantifiable learning?